Subgroups — C₂×#16838466 #21439722

Order 7644.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#3862	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838466	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×D₉₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₁₃×#3862	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇²×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
6	C₇×D₉₁	C₂×C₃	1	2	yes	complemented
7	C₂×ΓL(1,𝔽₈)⋊D₁₃	—	7	1	no	maximal
8	C₂×ΓL(1,𝔽₈)×D₁₃	—	7	1	no	maximal
9	C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
10	C₂×#64422	—	13	1	no	maximal, Hall
11	C₂×C₁₃×ΓL(1,𝔽₈)	—	7	1	no	—
12	C₂×C₁₃×ΓL(1,𝔽₈)	—	7	1	no	—
13	ΓL(1,𝔽₈)⋊D₁₃	—	7	2	no	—
14	ΓL(1,𝔽₈)×D₁₃	—	7	2	no	—
15	C₂×D₉₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	C₂×C₇×D₁₃	—	7	1	no	—
17	C₂×#3862	D₁₃	1	1	yes	characteristic, complemented
18	#64422	—	13	2	no	—
19	C₁₃×ΓL(1,𝔽₈)	—	7	1	no	—
20	C₁₃×ΓL(1,𝔽₈)	—	7	1	no	—
21	C₂×C₇×D₇	—	13	1	no	Hall
22	C₂×C₇×C₁₃	—	6	1	no	—
23	C₂×C₇×C₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
24	C₂×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
25	D₉₁	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
26	C₇×D₁₃	—	7	2	no	—
27	C₂×C₃×D₁₃	—	49	1	no	Hall
28	#3862	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
29	C₂×C₇²	C₃×D₁₃	1	1	yes	characteristic, complemented
30	C₇×D₇	—	13	2	no	—
31	C₇×C₁₃	—	6	1	no	—
32	C₇×C₁₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
33	C₇×C₁₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
34	C₂²×ΓL(1,𝔽₈)	—	91	1	no	—
35	C₂×AGL(1,𝔽₇)	—	91	1	no	—
36	C₂×C₃×C₁₃	—	49	1	no	—
37	C₃×D₁₃	—	49	2	no	—
38	C₂×D₁₃	—	7	1	no	Hall
39	C₇²	C₂×C₃×D₁₃	1	1	yes	characteristic, complemented, Sylow
40	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
41	C₂×ΓL(1,𝔽₈)	—	7	1	no	—
42	C₂×ΓL(1,𝔽₈)	—	91	2	no	—
43	AGL(1,𝔽₇)	—	91	2	no	—
44	C₃×C₁₃	—	49	1	no	Hall
45	C₂²×C₇	—	91	1	no	—
46	C₂×D₇	—	13	1	no	—
47	C₂×C₁₃	#64422	1	1	yes	characteristic, complemented
48	D₁₃	—	7	2	no	—
49	ΓL(1,𝔽₈)	—	7	1	no	—
50	ΓL(1,𝔽₈)	—	7	1	no	—
51	C₂×C₇	—	6	1	no	—
52	C₂×C₇	—	91	2	no	—
53	C₂×C₇	ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented
54	C₂×C₇	ΓL(1,𝔽₈)⋊D₁₃	1	1	yes	characteristic, complemented
55	D₇	—	13	2	no	—
56	C₁₃	C₂×#64422	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₃	—	637	1	no	Hall
58	C₇	—	6	1	no	—
59	C₇	C₂×ΓL(1,𝔽₈)×D₁₃	1	1	yes	characteristic, complemented
60	C₇	C₂×ΓL(1,𝔽₈)⋊D₁₃	1	1	yes	characteristic, complemented
61	C₂×C₃	—	49	1	no	—
62	C₂×C₃	—	637	2	no	—
63	C₂²	—	91	1	no	Sylow
64	C₃	—	49	1	no	Sylow
65	C₂	—	91	2	no	—
66	C₂	#16838466	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	#16838466
commutator, hypocenter	C₇²×C₁₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₁₃	C₂×C₃
LCS	G ▷ C₇²×C₁₃
UCS	C₁ ◁ C₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₁₃×#3862	C₂	semidirect
2. #16838466	C₂	direct (N × H)
3. C₂×C₇×D₉₁	C₃	semidirect
4. C₁₃×#3862	C₂²	semidirect
5. C₂×C₇²×C₁₃	C₂×C₃	semidirect
6. C₇×D₉₁	C₂×C₃	semidirect
9. C₇²×C₁₃	C₂²×C₃	semidirect
15. C₂×D₉₁	ΓL(1,𝔽₈)	semidirect
17. C₂×#3862	D₁₃	semidirect
23. C₂×C₇×C₁₃	AGL(1,𝔽₇)	semidirect
24. C₂×C₇×C₁₃	C₂×ΓL(1,𝔽₈)	semidirect
25. D₉₁	C₂×ΓL(1,𝔽₈)	semidirect
28. #3862	C₂×D₁₃	semidirect
29. C₂×C₇²	C₃×D₁₃	semidirect
32. C₇×C₁₃	C₂×AGL(1,𝔽₇)	semidirect
33. C₇×C₁₃	C₂²×ΓL(1,𝔽₈)	semidirect
39. C₇²	C₂×C₃×D₁₃	semidirect
47. C₂×C₁₃	#64422	semidirect
53. C₂×C₇	ΓL(1,𝔽₈)×D₁₃	semidirect
54. C₂×C₇	ΓL(1,𝔽₈)⋊D₁₃	semidirect
56. C₁₃	C₂×#64422	semidirect
59. C₇	C₂×ΓL(1,𝔽₈)×D₁₃	semidirect
60. C₇	C₂×ΓL(1,𝔽₈)⋊D₁₃	semidirect
66. C₂	#16838466	direct (N × H)

Subgroups — C2×#16838466 #21439722

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16838466 #21439722